How to sum the planar diagrams: A reformulation of U(N) lattice gauge theory, for N → ∞, in terms of a statistical ensemble of non-interacting random surfaces

Abstract We develop a theory of random surfaces on a lattice and derive loop equations in this theory that agree, apart from phase factors, with the loop equations that were found recently in U( N ) lattice gauge theory, in the limit of N → ∞. We then introduce extra phase factors in the ensemble of surfaces to exactly represent U(∞) gauge theory but we have not found a concise way of keeping track of these factors except for d = 3 dimensions. We also speculate on whether the extra phase factors could be rewritten in terms of anticommuting degrees of freedom.